Complex Exponential Function

The exponential function is defined by its differential equation exp'(z) = exp(z) with the initial value exp(0) = 1.

For no real number x ≠ 0 can exp(x) be computed in finitely many steps. All the numbers which our computers give us are only approximations of the true value of exp(x).

exp 1gauss 001
exp 1gauss 001

If Im(a) > 0 then the gridlines are spirals. The exponential function is periodic: exp(z + 2 π * i) = exp(z). One can see that in the range: For a = 1 the circular grid lines are about to close. The periodicity is also clear from the formula: exp(x + i * y) = exp(x) * (cos(y) + i * sin(y)) .

exp 2riemann 001

Each gridline intersects all the meridians with the same angle, such curves are called loxodromes.

exp 3ariemann 001